X-ray computed tomography (CT) has found extensive clinical applications in cancer, heart, and brain imaging. As CT has been increasingly used for cancer screening and pediatric imaging, there has arisen a push to reduce the radiation dose of clinical CT scans to become as low as reasonably achievable. Thus, iterative image reconstruction has been playing a more significant role in CT imaging. Iterative image reconstruction algorithms, as compared with traditional analytical algorithms, are promising in reducing the radiation dose while improving the CT image quality.
In X-ray computed tomography (CT), iterative reconstruction can be used to generate images. While there are various iterative reconstruction (IR) methods, such as the algebraic reconstruction technique, one common IR method is optimizing the expressionmin{∥Ax−p∥W2+βU(x)}to obtain the argument x that minimizes the expression. For example, in X-ray CT, A is the system matrix that represents X-ray trajectories (i.e., line integrals) along various rays from a source through an object OBJ to an X-ray detector (e.g., the X-ray transform corresponding to projections through the three-dimensional object OBJ onto a two-dimensional projection image p), p represents projection data for series of projection images taken at a various projection angles in a CT scan and corresponding (e.g., the projection data can be log-transform of the measured X-ray intensity at the X-ray detector), and x represents the reconstructed image of the X-ray attenuation of the object OBJ. The notation ∥g∥W2 signifies a weighted inner product of the form gTWg, wherein W is the weight matrix. For example, the weight matrix W can weigh the pixel values according to their noise statistics (e.g., the signal-to-noise ratio), in which case the weight matrix W is diagonal when the noise of each pixel is statistically independent. The data-matching term ∥Ax−p∥W2 is minimized when the forward projection A of the reconstructed image x provides a good approximation to all measured projection images p. In the above expression, U(x) is a regularization term, and β is a regularization parameter that weights the relative contributions of the data-matching term and the regularization term.
IR methods augmented with regularization can have several advantages over other reconstruction methods such as filtered back-projection. For example, IR methods augmented with regularization can produce high-quality reconstructions for few-view projection data and in the presence of significant noise. For few-view, limited-angle, and noisy projection scenarios, the application of regularization operators between reconstruction iterations seeks to tune the final and/or intermediate results to some a priori model. For example, enforcing positivity for the attenuation coefficients can provide a level of regularization based on the practical assumption that there are no regions in the object OBJ that cause an increase (i.e., gain) in the intensity of the X-ray radiation.
Other regularization terms can similarly rely on a priori knowledge of characteristics or constraints imposed on the reconstructed image. For example, minimizing the “total variation” (TV) in conjunction with projection on convex sets (POCS) is also a very popular regularization scheme. The TV-minimization algorithm assumes that the image is predominantly uniform over large regions with sharp transitions at the boundaries of the uniform regions, which is generally true for an image of a discrete number of organs, each with an approximately constant X-ray absorption coefficient (e.g., bone having a first absorption coefficient, the lungs having second coefficient, and the heart having a third coefficient). When the a priori model corresponds well to the image object OBJ, these regularized IR algorithms can produce impressive images, even though the reconstruction problem is significantly underdetermined (e.g., few-view scenarios), missing projection angles, or noisy. One advantage of using regularization is the flexibility of selecting from a wide range of regularization constraints.
A challenge of regularized IR algorithms is that they can require significantly more computational resources than filtered back-projection methods, often taking several hours to reach convergence for clinical CT scans using standard computing hardware. For certain quadratic regularizers, a one-step filtered back-projection algorithm can be used to reconstruct the reconstructed image of the Landweber algorithm for IR. This one-step filtered back-projection algorithm has the benefit of obtaining the result of the iterative Landweber algorithm, but much more quickly than performing IR. While the one-step filtered back-projection algorithm works for IR using a quadratic regularizer, no similar method is known to achieve the results of IR using FBP and a non-quadratic regularizer. Non-quadratic regularizers are known to advantageously achieve high levels of noise reduction while maintaining sharp edges in the reconstructed image. Further, non-quadratic regularizers can help achieve artifact suppression and better image quality than quadratic regularizers.